Yes, you can bypass all of this and put your non-reviewed pdfs on your personal website. No one will care, in part because no one will even know and if they do, they have to do the hard work of review in order to decide whether a paper is worth trusting. But how will anyone find out?
As an author, I've made a point never to publish in a paper not backed by a non-profit scientific society with a decent open-access policy, after doing so in the past. I also put all of my pdfs on my website including appropriate white papers that are working drafts, making it clear to others that they haven't been accepted through peer review.
As a reader, however, I keep up with the literature (aside from targeted Google Scholar searches) through Table of Contents emails from a few journals--mostly non-profit society journals but a few others including Nature. The editorial curation aspects which limit the number of accepted papers per issue make this possible. I'm not going to drink from the firehose of shit that is produced each month, especially before peer-review. Yeah, it sucks when your paper is accepted by reviewers but gets nixed by the editors because it's not cool enough, but... you know, get back to the bench and make it cooler if you want people to care. Welcome to the 21st century and attention scarcity.
You can change the name 'Nature' to whatever open-access journal you'd like, but the power behind it is that it's a sign that Important People have decided your paper is one of the 10 most impactful papers in your field that year. That's why people care.